Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 033, 8 pages      math-ph/0603011

A New Form of the Spherical Expansion of Zonal Functions and Fourier Transforms of SO(d)-Finite Functions

Agata Bezubik a and Aleksander Strasburger b
a) Institute of Mathematics, University of Bialystok, Akademicka 2, 15-267 Bialystok, Poland
b) Department of Econometrics and Informatics, Warsaw Agricultural University, Nowoursynowska 166, 02-787 Warszawa, Poland

Received November 30, 2005, in final form February 17, 2006; Published online March 03, 2006

This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for problems involving Fourier transforms of functions with rotational symmetry. The method used to derive the expansion formula is based entirely on differential methods and completely avoids the use of various integral identities commonly used in this context. Some new identities for the Fourier transform are derived and as a byproduct seemingly new recurrence relations for the classical Bessel functions are obtained.

Key words: spherical harmonics; zonal harmonic polynomials; Fourier-Laplace expansions; special orthogonal group; Bessel functions; Fourier transform; Bochner identity.

pdf (202 kb)   ps (154 kb)   tex (11 kb)


  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge, Cambridge University Press, 1999.
  2. Bezubik A., Dabrowska A., Strasburger A., On the Fourier transform of SO(d)-finite measures on the unit sphere, Arch. Math. (Basel), 2005, V.84, 470-480.
  3. Bezubik A., Dabrowska A., Strasburger A., On spherical expansions of zonal functions on Euclidean spheres, submitted.
  4. zu Castell W., Filbir F., Radial Basis functions and corresponding zonal series expansions on the sphere, J. Approx. Theory, 2005, V.134, 65-79.
  5. Faraut J., Analyse harmonique et fonctions speciales, in Deux Cours d'Analyse Harmonique, Editors J. Faraut and K. Harzallah, Ecole d'Ètè d'Analyse Harmonique de Tunis, 1984, Basel, Birkhäuser Verlag, 1987.
  6. Gonzalez Vieli F.J., Inversion de Fourier ponctuelle des distributions à support compact, Arch. Math. (Basel), 2000, V.75, 290-298.
  7. Lucquiaud J.C., Generalization sous forme covariante des polynomes de Gegenbauer, J. Math. Pures Appl. (9), 1984, V.63, 265-282.
  8. Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics, 3rd ed., Berlin, Springer-Verlag, 1966.
  9. Müller C., Analysis of spherical symmetries in Euclidean spaces, New York, Springer-Verlag, 1998.
  10. Stein E.M., Weiss G., Introduction to harmonic analysis on Euclidean spaces, Princeton, Princeton University Press, 1971.
  11. Strasburger A., A generalization of the Bochner identity, Exposition. Math., 1993, V.11, 153-157.
  12. Vilenkin N.J., Special functions and the theory of group representations, Moscow, Nauka, 1965 (in Russian).
  13. Wawrzynczyk A., Group representations and special functions, Dordrecht - Warszawa, D. Reidel and PWN, 1984.

Previous article   Next article   Contents of Volume 2 (2006)